Question 1

An equation for the tangent line to the curve $y = \left( x ^ { 5 } + x - 1 \right) ^ { 5 }$ at the point where x = 1 is

Accepted Answer

A) y = 30x 
B) y = 5x + 1 
C) y = 30x - 1 
D) y = 30x - 29 
A) y = 30x 
B) y = 5x + 1 
C) y = 30x - 1 
D) y = 30x - 29 D

Question 2

If $f ( x ) = \left( x + 7 e ^ { - x } \right) ^ { 2 }$ ,then $f ^ { \prime } ( x ) = x + 7 e ^ { - x }$ .

Accepted Answer

A) True 
 B)False  True

Question 3

A company makes a certain product for $4 each and sells it for $8.If the company has overhead expenses of $10,000 per year,how many of its products must be made and sold to break even?

Accepted Answer

A) 20,000 
B) 40,000 
C) 2,500 
D) 10,000 
A) 20,000 
B) 40,000 
C) 2,500 
D) 10,000 C

Question 4

The second derivative test reveals that $f ( x ) = x ^ { 4 } - 4 x ^ { 2 } + 1$ has

Accepted Answer

A) neither a relative maximum nor a relative minimum. 
B) a relative maximum and two relative minima. 
C) a relative maximum only. 
D) a relative minimum only. 
A) neither a relative maximum nor a relative minimum. 
B) a relative maximum and two relative minima. 
C) a relative maximum only. 
D) a relative minimum only.

Question 5

Evaluate $\int x \sqrt { x ^ { 2 } + 2 } d x$ .

Accepted Answer

A)  $\left( x ^ { 2 } + 2 \right) ^ { 3 / 2 } + C$ 
B)  $\frac { x ^ { 3 } } { 3 } + 2 x + C$ 
C)  $\frac { 3 \left( x ^ { 2 } + 2 \right) ^ { 3 / 2 } } { 4 } + C$ 
D)  $\frac { \left( x ^ { 2 } + 2 \right) ^ { 3 / 2 } } { 3 } + C$ 
A)  $\left( x ^ { 2 } + 2 \right) ^ { 3 / 2 } + C$ 
B)  $\frac { x ^ { 3 } } { 3 } + 2 x + C$ 
C)  $\frac { 3 \left( x ^ { 2 } + 2 \right) ^ { 3 / 2 } } { 4 } + C$ 
D)  $\frac { \left( x ^ { 2 } + 2 \right) ^ { 3 / 2 } } { 3 } + C$

Question 6

Differentiate: $f ( x ) = x ^ { 6 } + 7$

Accepted Answer

A)  $6 x ^ { 7 } + 7 x$ 
B)  $5 x ^ { 5 }$ 
C)  $6 x ^ { 5 } + 7$ 
D)  $6 x ^ { 5 }$ 
A)  $6 x ^ { 7 } + 7 x$ 
B)  $5 x ^ { 5 }$ 
C)  $6 x ^ { 5 } + 7$ 
D)  $6 x ^ { 5 }$

Question 7

Money is transferred continuously into an account at the constant rate of $1,000 per year.Assume the account earns interest at the annual rate of 8% compounded continuously.Compute the future value of the income stream over a 5 year period.

Accepted Answer

A) $12,295.62 
B) $6,147.81 
C) $491.82 
D) $24,591.23 
A) $12,295.62 
B) $6,147.81 
C) $491.82 
D) $24,591.23

Question 8

Solve the given equation for x. $6 = 5 + 9 e ^ { 3 x }$

Accepted Answer

A)  $\frac { e ^ { 9 } } { 3 }$ 
B)  $\frac { \ln 9 } { 3 }$ 
C)  $- \frac { \ln 9 } { 3 }$ 
D)  $3 e ^ { 9 }$ 
A)  $\frac { e ^ { 9 } } { 3 }$ 
B)  $\frac { \ln 9 } { 3 }$ 
C)  $- \frac { \ln 9 } { 3 }$ 
D)  $3 e ^ { 9 }$

Question 9

Find the indicated limit if it exists. $\lim _ { x \rightarrow 4 } \frac { \sqrt { x } - 2 } { x - 4 }$

Accepted Answer

A)  $- \frac { 1 } { 4 }$ 
B) Does not exist 
C)  $\frac { 1 } { 4 }$ 
D) 4 
A)  $- \frac { 1 } { 4 }$ 
B) Does not exist 
C)  $\frac { 1 } { 4 }$ 
D) 4

Question 10

The function y = ln 6x is concave downward everywhere.

Accepted Answer

A) True 
 B)False

Question 11

Find the intervals of increase and decrease for $f ( x ) = \frac { 8 x - 1 } { - 2 x + 10 }$ .Round numbers to two decimal places,if necessary.

Accepted Answer

A) Increasing on x  0.12 and on x > 5,decreasing on 0.12  5 
C) Increasing on x  5 
D) Increasing on x  5 
A) Increasing on x  0.12 and on x > 5,decreasing on 0.12  5 
C) Increasing on x  5 
D) Increasing on x  5

Question 12

$\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 2 } { x } d x = \frac { x ^ { 3 } } { 3 } - \frac { 3 x ^ { 2 } } { 2 } + 2 \ln | x | + C$

Accepted Answer

A) True 
 B)False

Question 13

Find the absolute maximum of the function $f ( x ) = x ^ { 3 }$ on the interval $- \frac { 1 } { 2 } \leq x \leq 1$ .

Accepted Answer

A) 1 
B) -1 
C) 0 
D)  $\frac { 1 } { 8 }$ 
A) 1 
B) -1 
C) 0 
D)  $\frac { 1 } { 8 }$

Question 14

Find the equation of the tangent line to the graph of $f ( x ) = \frac { 3 } { x }$ at the point $\left( 4 , \frac { 3 } { 4 } \right)$ .

Accepted Answer

A)  $y = \frac { 3 } { 4 } x + \frac { 3 } { 2 }$ 
B)  $y = - \frac { 3 } { 16 } x + \frac { 3 } { 2 }$ 
C)  $y = - \frac { 3 } { 4 } x + \frac { 3 } { 2 }$ 
D)  $y = \frac { 3 } { 16 } x$ 
A)  $y = \frac { 3 } { 4 } x + \frac { 3 } { 2 }$ 
B)  $y = - \frac { 3 } { 16 } x + \frac { 3 } { 2 }$ 
C)  $y = - \frac { 3 } { 4 } x + \frac { 3 } { 2 }$ 
D)  $y = \frac { 3 } { 16 } x$

Question 15

Find the equation of the tangent line to $f ( x ) = 4 \ln x ^ { 3 }$ at x = e.

Accepted Answer

A)  $y = \frac { 12 } { e } x + 24$ 
B) y = 12x 
C)  $y = \frac { 12 } { e } x$ 
D)  $y = \frac { 12 } { e } x - 24$ 
A)  $y = \frac { 12 } { e } x + 24$ 
B) y = 12x 
C)  $y = \frac { 12 } { e } x$ 
D)  $y = \frac { 12 } { e } x - 24$

Question 16

The owner of an appliance store expects to sell 500 toasters this year.Each toaster costs her $6 dollars to purchase,and each time she orders a shipment of toasters,it costs $30.In addition,it costs $4 a year to store each toaster.Assuming the toasters sell out at a uniform rate and that the owner never allows herself to run out of toasters,how many toasters should be ordered in each shipment to minimize the annual cost? (Round any fractional amounts.)

Accepted Answer

A) 87 
B) 22 
C) 44 
D) 33 
A) 87 
B) 22 
C) 44 
D) 33

Question 17

Determine where the graph of $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x + 1$ is concave down.

Accepted Answer

A) x > -1 
B) x  1 
A) x > -1 
B) x  1

Question 18

Find the slope (if possible)of the line that passes through the given pair of points. (7,0)and (17,19)

Accepted Answer

A)  $\frac { 10 } { 19 }$ 
B)  $- \frac { 19 } { 10 }$ 
C)  $\frac { 19 } { 10 }$ 
D)  $- \frac { 10 } { 19 }$ 
A)  $\frac { 10 } { 19 }$ 
B)  $- \frac { 19 } { 10 }$ 
C)  $\frac { 19 } { 10 }$ 
D)  $- \frac { 10 } { 19 }$

Question 19

Find the points of intersection (if any)of the given pair of curves. y = x + 8 and y = 2x + 4

Accepted Answer

A) (12,20) 
B) (4,12) 
C) (1,-4) 
D) (0,6) 
A) (12,20) 
B) (4,12) 
C) (1,-4) 
D) (0,6)

Question 20

An efficiency study of the morning shift at a certain factory indicates that an average worker who arrives on the job at 8:00 A.M.will have assembled $f ( x ) = - x ^ { 3 } + 5 x ^ { 2 } + 16 x$ transistor radios x hours later.How many radios will such a worker assemble between 10:00 and 11:00 A.M.?

Accepted Answer

A) 18 
B) 20 
C) 22 
D) 15 
A) 18 
B) 20 
C) 22 
D) 15

An equation for the tangent line to the curve $y = \left( x ^ { 5 } + x - 1 \right) ^ { 5 }$ at the point where x = 1 is

If $f ( x ) = \left( x + 7 e ^ { - x } \right) ^ { 2 }$ ,then $f ^ { \prime } ( x ) = x + 7 e ^ { - x }$ .

A company makes a certain product for $4 each and sells it for $8.If the company has overhead expenses of $10,000 per year,how many of its products must be made and sold to break even?

The second derivative test reveals that $f ( x ) = x ^ { 4 } - 4 x ^ { 2 } + 1$ has

Evaluate $\int x \sqrt { x ^ { 2 } + 2 } d x$ .

Differentiate: $f ( x ) = x ^ { 6 } + 7$

Money is transferred continuously into an account at the constant rate of $1,000 per year.Assume the account earns interest at the annual rate of 8% compounded continuously.Compute the future value of the income stream over a 5 year period.

Solve the given equation for x. $6 = 5 + 9 e ^ { 3 x }$

Find the indicated limit if it exists. $\lim _ { x \rightarrow 4 } \frac { \sqrt { x } - 2 } { x - 4 }$

The function y = ln 6x is concave downward everywhere.

Find the intervals of increase and decrease for $f ( x ) = \frac { 8 x - 1 } { - 2 x + 10 }$ .Round numbers to two decimal places,if necessary.

$\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 2 } { x } d x = \frac { x ^ { 3 } } { 3 } - \frac { 3 x ^ { 2 } } { 2 } + 2 \ln | x | + C$

Find the absolute maximum of the function $f ( x ) = x ^ { 3 }$ on the interval $- \frac { 1 } { 2 } \leq x \leq 1$ .

Find the equation of the tangent line to the graph of $f ( x ) = \frac { 3 } { x }$ at the point $\left( 4 , \frac { 3 } { 4 } \right)$ .

Find the equation of the tangent line to $f ( x ) = 4 \ln x ^ { 3 }$ at x = e.

Determine where the graph of $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x + 1$ is concave down.

Find the slope (if possible)of the line that passes through the given pair of points. (7,0)and (17,19)

Find the points of intersection (if any)of the given pair of curves. y = x + 8 and y = 2x + 4

An efficiency study of the morning shift at a certain factory indicates that an average worker who arrives on the job at 8:00 A.M.will have assembled $f ( x ) = - x ^ { 3 } + 5 x ^ { 2 } + 16 x$ transistor radios x hours later.How many radios will such a worker assemble between 10:00 and 11:00 A.M.?

Functions,graphs,and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Itegration

Additional Topics in Integration

Calculus of Several Variables

Differential Equations

Infinite Series and Taylor Series Approximations

Probability and Calculus

Trigonometric Functions

Mathematical Problem Set: Calculus, Algebra, and Optimization

Filters

Exam 12: Calculus Practice Questions

An equation for the tangent line to the curve y=(x5+x−1)5y = \left( x ^ { 5 } + x - 1 \right) ^ { 5 }y=(x5+x−1)5 at the point where x = 1 is

If f(x)=(x+7e−x)2f ( x ) = \left( x + 7 e ^ { - x } \right) ^ { 2 }f(x)=(x+7e−x)2 ,then f′(x)=x+7e−xf ^ { \prime } ( x ) = x + 7 e ^ { - x }f′(x)=x+7e−x .

A company makes a certain product for $4 each and sells it for $8.If the company has overhead expenses of $10,000 per year,how many of its products must be made and sold to break even?

The second derivative test reveals that f(x)=x4−4x2+1f ( x ) = x ^ { 4 } - 4 x ^ { 2 } + 1f(x)=x4−4x2+1 has

Evaluate ∫xx2+2dx\int x \sqrt { x ^ { 2 } + 2 } d x∫xx2+2​dx .

Differentiate: f(x)=x6+7f ( x ) = x ^ { 6 } + 7f(x)=x6+7

Money is transferred continuously into an account at the constant rate of $1,000 per year.Assume the account earns interest at the annual rate of 8% compounded continuously.Compute the future value of the income stream over a 5 year period.

Solve the given equation for x. 6=5+9e3x6 = 5 + 9 e ^ { 3 x }6=5+9e3x

Find the indicated limit if it exists. lim⁡x→4x−2x−4\lim _ { x \rightarrow 4 } \frac { \sqrt { x } - 2 } { x - 4 }limx→4​x−4x​−2​

The function y = ln 6x is concave downward everywhere.

Find the intervals of increase and decrease for f(x)=8x−1−2x+10f ( x ) = \frac { 8 x - 1 } { - 2 x + 10 }f(x)=−2x+108x−1​ .Round numbers to two decimal places,if necessary.

∫x3−3x2+2xdx=x33−3x22+2ln⁡∣x∣+C\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 2 } { x } d x = \frac { x ^ { 3 } } { 3 } - \frac { 3 x ^ { 2 } } { 2 } + 2 \ln | x | + C∫xx3−3x2+2​dx=3x3​−23x2​+2ln∣x∣+C

Find the absolute maximum of the function f(x)=x3f ( x ) = x ^ { 3 }f(x)=x3 on the interval −12≤x≤1- \frac { 1 } { 2 } \leq x \leq 1−21​≤x≤1 .

Find the equation of the tangent line to the graph of f(x)=3xf ( x ) = \frac { 3 } { x }f(x)=x3​ at the point (4,34)\left( 4 , \frac { 3 } { 4 } \right)(4,43​) .

Find the equation of the tangent line to f(x)=4ln⁡x3f ( x ) = 4 \ln x ^ { 3 }f(x)=4lnx3 at x = e.

Determine where the graph of f(x)=x3−3x2−9x+1f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x + 1f(x)=x3−3x2−9x+1 is concave down.

Find the slope (if possible)of the line that passes through the given pair of points. (7,0)and (17,19)

Find the points of intersection (if any)of the given pair of curves. y = x + 8 and y = 2x + 4

Functions,graphs,and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Itegration

Additional Topics in Integration

Calculus of Several Variables

Differential Equations

Infinite Series and Taylor Series Approximations

Probability and Calculus

Trigonometric Functions

Mathematical Problem Set: Calculus, Algebra, and Optimization

Filters

An equation for the tangent line to the curve $y = \left( x ^ { 5 } + x - 1 \right) ^ { 5 }$ at the point where x = 1 is

If $f ( x ) = \left( x + 7 e ^ { - x } \right) ^ { 2 }$ ,then $f ^ { \prime } ( x ) = x + 7 e ^ { - x }$ .

The second derivative test reveals that $f ( x ) = x ^ { 4 } - 4 x ^ { 2 } + 1$ has

Evaluate $\int x \sqrt { x ^ { 2 } + 2 } d x$ .

Differentiate: $f ( x ) = x ^ { 6 } + 7$

Solve the given equation for x. $6 = 5 + 9 e ^ { 3 x }$

Find the indicated limit if it exists. $\lim _ { x \rightarrow 4 } \frac { \sqrt { x } - 2 } { x - 4 }$

Find the intervals of increase and decrease for $f ( x ) = \frac { 8 x - 1 } { - 2 x + 10 }$ .Round numbers to two decimal places,if necessary.

$\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 2 } { x } d x = \frac { x ^ { 3 } } { 3 } - \frac { 3 x ^ { 2 } } { 2 } + 2 \ln | x | + C$

Find the absolute maximum of the function $f ( x ) = x ^ { 3 }$ on the interval $- \frac { 1 } { 2 } \leq x \leq 1$ .

Find the equation of the tangent line to the graph of $f ( x ) = \frac { 3 } { x }$ at the point $\left( 4 , \frac { 3 } { 4 } \right)$ .

Find the equation of the tangent line to $f ( x ) = 4 \ln x ^ { 3 }$ at x = e.

Determine where the graph of $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x + 1$ is concave down.